Discarding to opponent's crib

Let's continue our look at the discarding tables compiled by Hessel, Colvert and Rasmussen by examining the discard to opponent's crib. This is usually a more difficult proposition than discarding to your own crib, since you must balance two contradictory goals: maximizing the value of your hand and minimizing the value of dealer's crib. A good way to develop a feel for this is to study the following tables, which give the average value of the crib for each of the possible two-card tosses.

The methodologies behind these tables were described in the previous article, so I'll go straight to the data. Naturally, since we're discarding to opponent's crib, lower numbers are better.

Discarding to opponent's crib (Hessel)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 6.07 5.07 5.17 5.74 6.06 4.93 4.95 4.92 4.66 4.46 4.72 4.41 4.34
2 5.07 6.43 7.34 5.44 6.17 5.13 5.12 5.03 4.82 4.64 4.91 4.60 4.53
3 5.17 7.34 6.78 6.10 6.85 4.92 5.16 5.08 4.82 4.70 4.97 4.66 4.59
4 5.74 5.44 6.10 6.59 7.46 5.47 4.91 5.02 4.75 4.55 4.80 4.49 4.43
5 6.06 6.17 6.85 7.46 9.39 7.66 7.08 6.36 6.22 7.46 7.75 7.42 7.31
6 4.93 5.13 4.92 5.47 7.66 7.17 6.64 6.05 6.31 4.41 4.61 4.29 4.25
7 4.95 5.12 5.16 4.91 7.08 6.64 7.25 7.88 5.46 4.44 4.73 4.44 4.38
8 4.92 5.03 5.08 5.02 6.36 6.05 7.88 6.76 5.97 5.02 4.65 4.38 4.31
9 4.66 4.82 4.82 4.75 6.22 6.31 5.46 5.97 6.44 5.52 4.98 4.14 4.13
10 4.46 4.64 4.70 4.55 7.46 4.41 4.44 5.02 5.52 6.11 5.60 4.65 3.99
J 4.72 4.91 4.97 4.80 7.75 4.61 4.73 4.65 4.98 5.60 6.56 5.55 4.89
Q 4.41 4.60 4.66 4.49 7.42 4.29 4.44 4.38 4.14 4.65 5.55 5.89 4.56
K 4.34 4.53 4.59 4.43 7.31 4.25 4.38 4.31 4.13 3.99 4.89 4.56 5.72

Discarding to opponent's crib (Colvert)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 6.2 5.0 5.1 5.7 6.0 4.9 4.9 4.8 4.6 4.4 4.7 4.4 4.3
2 5.0 6.4 7.3 5.3 6.1 5.0 5.0 4.9 4.8 4.6 4.8 4.5 4.4
3 5.1 7.3 6.8 5.9 6.7 4.9 5.0 5.0 4.8 4.6 4.9 4.5 4.4
4 5.7 5.3 5.9 6.6 7.2 5.3 4.8 4.9 4.7 4.5 4.7 4.4 4.3
5 6.0 6.1 6.7 7.2 9.3 7.4 6.9 6.2 6.1 7.4 7.6 7.3 7.2
6 4.9 5.0 4.9 5.3 7.4 7.0 6.4 5.7 6.3 4.3 4.5 4.2 4.1
7 4.9 5.0 5.0 4.8 6.9 6.4 7.1 7.8 5.2 4.3 4.7 4.3 4.2
8 4.8 4.9 5.0 4.9 6.2 5.7 7.8 6.6 5.7 4.9 4.6 4.3 4.2
9 4.6 4.8 4.8 4.7 6.1 6.3 5.2 5.7 6.3 5.4 5.0 4.1 4.0
10 4.4 4.6 4.6 4.5 7.4 4.3 4.3 4.9 5.4 6.0 5.4 4.5 3.8
J 4.7 4.8 4.9 4.7 7.6 4.5 4.7 4.6 5.0 5.4 6.5 5.4 4.7
Q 4.4 4.5 4.5 4.4 7.3 4.2 4.3 4.3 4.1 4.5 5.4 5.8 4.4
K 4.3 4.4 4.4 4.3 7.2 4.1 4.2 4.2 4.0 3.8 4.7 4.4 5.6

Discarding to opponent's crib (Rasmussen)

  A 2 3 4 5 6 7 8 9 10 J Q K
A 5.59 5.17 4.96 5.62 5.81 4.97 4.81 4.84 4.34 4.54 4.64 4.24 4.33
2 5.17 6.19 7.52 5.21 5.79 4.79 4.80 4.90 4.57 4.54 4.61 4.58 4.45
3 4.95 7.52 6.11 5.74 6.72 4.81 4.85 5.20 5.18 4.58 4.71 4.61 4.43
4 5.61 5.20 5.74 6.00 6.44 5.06 5.00 4.94 4.57 4.58 5.14 4.50 4.36
5 5.81 5.79 6.72 6.43 9.09 6.87 7.08 6.39 6.06 7.22 8.14 7.10 7.13
6 4.96 4.79 4.81 5.05 6.86 6.30 6.18 5.86 6.20 4.22 4.53 4.14 4.08
7 4.81 4.80 4.84 4.99 7.08 6.17 6.93 6.67 5.10 4.17 4.69 4.24 4.25
8 4.84 4.90 5.19 4.93 6.39 5.86 6.67 7.91 5.89 5.59 4.58 4.30 4.15
9 4.33 4.57 5.17 4.57 6.06 6.20 5.10 5.89 6.52 5.30 4.86 4.12 3.94
10 4.54 4.53 4.57 4.57 7.21 4.22 4.17 5.58 5.29 6.19 5.95 4.64 3.85
J 4.64 4.61 4.70 5.14 8.13 4.53 4.69 4.57 4.86 5.95 5.64 5.46 4.63
Q 4.23 4.57 4.61 4.50 7.10 4.14 4.24 4.29 4.11 4.63 5.46 5.36 4.52
K 4.33 4.45 4.43 4.36 7.12 4.07 4.24 4.15 3.93 3.84 4.62 4.51 5.59

Once again, the Hessel and Colvert figures match each other closely, the most notable disparities concerning mid-card tosses such as 6-7, 6-8 and 7-9, to which Hessel gives higher values than Colvert. Rasmussen's averages, which closely matched the others for discarding to your own crib, occasionally differ markedly for discarding to opponent's crib. Most of the departures concern dangerous tosses such as 7-8 or 8-8. Since Ras's figures are based on his own games, there are far fewer samples for these tosses than for less dangerous (and thus more common) ones. For example, in 87,111 tabulated discards to his opponents' cribs, the 10-K toss occurred 3,884 times, the 5-5 toss only 58 times. This results in a disproportionately large statistical margin of error for 5-5 and other infrequent tosses, and this probably accounts for most of the difference between Ras's figures and the others.

In fact, for the most frequent discards (those averaging under five points in opponent's crib), the three tables match each other quite closely. The only significant exception is A-9, which Ras reckons gives up 4.34 points, 0.3 less than Hessel and Colvert.

Here is a summary of notable differences between Rasmussen's figures and the others:

Toss:

Average crib:

Rasmussen Colvert (Diff.) Hessel (Diff.)
5-J 8.14 7.6 +0.5  7.75 +0.4 
8-8 7.91 6.6 +1.3  6.76 +1.1 
5-6 6.87 7.4 -0.5  7.66 -0.8 
7-8 6.67 7.8 -1.1  7.88 -1.2 
4-5 6.44 7.2 -0.8  7.46 -1.0 
6-6 6.30 7.0 -0.7  7.17 -0.9 
6-7 6.18 6.3 -0.1  6.64 -0.5 
3-3 6.11 6.8 -0.7  6.78 -0.7 
4-4 6.00 6.6 -0.6  6.59 -0.6 
10-J 5.95 5.4 +0.6  5.60 +0.4 
J-J 5.64 6.5 -0.9  6.56 -0.9 
A-A 5.59 6.2 -0.6  6.07 -0.5 
8-10 5.59 4.9 +0.7  5.02 +0.6 
Q-Q 5.36 5.8 -0.4  5.89 -0.5 
3-9 5.18 4.8 +0.4  4.82 +0.4 
4-J 5.14 4.7 +0.4  4.80 +0.3 
4-6 5.06 5.3 -0.2  5.47 -0.4 
A-9 4.34 4.6 -0.3  4.66 -0.3 

As with the dealer numbers, I have distilled these three pone discard tables into a single set of figures for use in my own personal analysis and over-the-board decision-making. Click here to see them.

The tables reinforce most of the conventional wisdom about discarding as pone. Nevertheless, there are a few surprises.

Clearly the most dangerous cards to toss are 5s and two-card combinations totaling five or fifteen. Somewhat less dangerous are mid-card pairs, followed by low pairs and then high pairs. Since all of these combinations are worth two points going in (remember, any five-card hand containing an A-4, 2-3 or 5 must be worth at least two points), the danger of tossing any of them to your opponent's crib is obvious.

Less obvious is the danger of tossing two mid-cards that don't form a pair or 15. Note, for example, how the 6-7 discard gives up more points than A-A, A-4, A-5, 2-5, 5-9, Q-Q and K-K, even though the latter tosses are worth two going in! Likewise, 6-8 and 8-9 are riskier than A-4 and K-K, and Hessel and Ras give 3-4 as more dangerous than A-A, A-4, Q-Q and K-K. As noted last time, touching cards derive more added value in the crib than pairs, which have relatively few possibilities for improvement.

Touching cards are much safer to throw if they are edge cards that can only be extended into a run in one direction. These are about as safe as near cards (cards separated by one rank) which form a run only when combined with an "inside" card. Thus the 3-4 toss (touching cards), which can form a run with either a 2 or a 5, is more dangerous than A-2 (edge cards) or A-3 (near cards). Likewise, 9-10, 10-J and J-Q give up about a point more than 10-Q, J-K and Q-K. In fact, the latter three tosses are surprisingly safe, often preferable to tossing a wider, but lower, combination such as 3-8 or 4-7.

As you would expect, tossing a J adds roughly  point to the value of the crib, due to the possibility of His Nobs. If you toss an A, 2, 3 or 4 to dealer's crib, it's better to toss a 9, 10, Q or K with it, instead of a J. If you must split a pair of Js, toss the J from the longer suit. Thus, from this hand:

A 5 10 J J Q

toss A J, not A J. The J will fetch one point on ten different cuts (since you were dealt three diamonds). The J will fetch one point on 12 cuts.

In the above example, note that the J toss is statistically correct even though it means risking a crib flush. Your chance of giving up this rarest of cribbage scores is about 1 in 140 when you toss two cards of the same suit. This represents an added risk of approximately .036 points . If you tossed the J instead, you would add .043 points  to the average value of the crib while subtracting the same value from your hand, for a deficit of .086 points. Granted, it's a subtlety, but then, cribbage is a subtle game. DeLynn Colvert discusses J discards on p. 50 of Play Winning Cribbage (Third Edition).

Note that the 1 in 140 chance of giving up a crib flush on a same-suit discard as pone is a tad higher than the corresponding 1 in 150 chance of getting a crib flush as dealer (see How to analyze discards, part 1). This is because a same-suit discard more likely to be made by dealer than by pone.

What are the best tosses to your opponent's crib? Wide cards obviously, but which ones? It turns out that tossing a mid-card and a high card is safer than tossing a low card and a high card. This is because a combination such as 8-K cannot possibly be combined into a single score, unlike a combo such as A-K, which can make a 15 with the addition of a 4. Either alternative is safer than tossing a low card and a mid-card.

The single safest card to throw your opponent has long been assumed to be the K, and this is borne out by Hessel's and Colvert's tables. However, Rasmussen's statistics often give the edge to the Q instead. For example, Ras has K-K giving up more points on average than Q-Q, while in Colvert and Hessel's figures, the reverse is true. Have Ras's figures been distorted by the statistical margin of error in his samples? Perhaps, but a more likely explanation is cribbage psychology.

Many human players, especially experienced ones, will actually toss a lone K to their own crib in preference to a lone Q, under the assumption that a K is more likely to be paired by pone's toss. Some cribbage textbooks even advocate this explicitly: see p. 37 of Colvert's book, where he advises tossing 2-K (not 2-10) to your own crib from 2-5-10-J-Q-K. Wergin recommends the exact same toss from this hand on p. 90 of Win at Cribbage. If your opponent follows this advice, you will probably give up fewer pairs, over the long run, by tossing her a Q instead of a K. However, since the K is less likely to become part of a run, you'll give up fewer runs with the K. This is probably why Rasmussen's statistics give 9-Q a better chance of producing a bust crib than 9-K, even though the latter toss gives up fewer average points.

The Q/K debate is an effective demonstration of the human bias in Rasmussen's figures, which are based on real games, compared with the computer bias of Hessel's and Colvert's figures, which are based on software simulations.

Here is a compendium of the safest tosses to opponent's crib, based on Rasmussen's numbers. In most cases, you are concerned with giving up the fewest average points, and the tosses on the left do just that. 10-K is best, followed by 9-K, 6-K, 9-Q and so on. If you are playing desperation defense, try one of the tosses in the middle list. These have the best chance of holding the crib to two points or less. But some of them are risky: discards such as 10-Q, 10-K and Q-K tend to be all-or-nothing affairs, producing lots of bust cribs, but lots of barnburners as well. Finally, if you're protecting a large lead, you might try one of the tosses on the right, which are the least likely to give up a crib worth eight or more points.

Optimal discards to opponent's crib (Rasmussen)

Toss:

Average crib:

Toss:

Cribs worth 2 or less:

Toss:

Cribs worth 8 or more:

10-K 3.85 10-K 45.5% 9-Q 11.5%
9-K 3.94 Q-K 42.4% 9-K 11.6%
6-K 4.08 9-Q 42.2% 7-10 12.7%
9-Q 4.12 6-K 41.4% 6-K 13.1%
6-Q 4.14 10-Q 41.0% 8-Q 13.1%
8-K 4.15 6-Q 40.0% A-3 13.1%
7-10 4.17 4-Q 39.6% 8-K 13.2%
6-10 4.22 4-K 39.5% 6-10 14.2%
A-Q 4.24 6-10 39.1% 7-K 14.3%
7-Q

4.24

8-K 38.3% 3-7 14.4%
7-K 4.25 7-K 38.3% 10-K 14.4%
8-Q 4.30 9-K 38.1% 2-J 14.5%
A-K 4.33 4-10 38.1% 6-Q 14.7%

Some tosses appear in only one list, others appear in all three, but in different places. This demonstrates that the specific toss you make to your opponent's crib should be dictated by the requirements of your board position, and that the same hand may be played differently depending on the score. Suppose you're dealt:

A-3-7-9-10-K

What do you toss if the score is?

  1. 90-90*
     
  2. 105-86*
     
  3. 105-98*
     
  4. 114-111*

(remember, the asterisks indicate the player dealing)

Answers:

  1. 90-90*
    You need to score at least a half dozen points here to get into good position (96+ points) for your three counts. Keeping A-3-9-10 will get you the most points, on average, of any four-card combination available to you. It's also a decent pegging hand, with two low cards and two higher touching cards. Although the resulting 7-K toss is not as safe as 9-K or 10-K, it's safe enough for this situation, since dealer is also six points short of the positional hole (96).
     
  2. 105-86*
    Your lousy hand gives you no chance of going out this deal, so forget about trying for a skunk. You must maximize your chances of going out on the three consecutive counts you'll get starting next deal (your hand and crib as dealer, then your hand as pone on the following deal). You already have 105 points, so this shouldn't be too hard, provided your opponent doesn't go out first. Since she's only at 86, you should be playing the cribbage equivalent of a prevent defense. This means sacrificing a point here or there to prevent a major catastrophe, such as a barnburner crib. Toss the combination with the least chance of giving up an 8+ point crib: the 9-K.
     
    Yes, 10-K gives up the fewest points on average, but as Rasmussen points out, it's an all-or-nothing toss. If it doesn't catch a 5, you have an excellent chance of shutting out your opponent's crib. But if it does catch a 5 (in the cut or dealer's discard), you're in deep trouble. Indeed, the 10-K toss can give up as many as 20 points (if the other three cards are all 5s), exactly the kind of ass-kicker crib that can turn an easy win into a loss. The worst you could do with 9-K is to give up 14 points (again, with three 5s). That still isn't great, but it's a lot more survivable in this position.
     
  3. 105-98*
    Again, you're at 105 points, and you know you can't go out this deal with the cards you were dealt. Unfortunately, dealer is at 98 points, close enough to go out on her three counts with average cards. Your only hope is to slow her down so that she doesn't go out as pone next deal. If that happens, you'll be a heavy favorite to go out on your three counts, even if you don't score a single point here. Toss 10-K and take your best shot at shutting out her crib.
     
  4. 114-111*
    If you don't go out with this hand, dealer almost certainly will, regardless of what you toss. So keep A-3-9-10, retaining your only shot at the seven points you need (on a 2 cut). If you cut an A, 3 or 5 instead, you'll get four points — enough to win if you also manage to peg three (say, by getting a pair and a go).

If you're unfamiliar with things like positional holes and three counts, then I recommend reading the Cribbage for the Expert chapter of Colvert's Play Winning Cribbage, and Part II of Chamber's Cribbage: A New Concept. You can view excerpts from both books at the ACC's online Cribbage Tip Library.

A final thought: the average crib given up by Rasmussen is 4.68 points, quite a bit lower than the 4.84 points he gets in his own cribs. If you project that spread over the nine deals of an average game, it's an advantage of .72 points per game over his opponents. That might not seem like much, but even a one point spread per game translates into a 2% increase in winning percentage (Colvert, p. 114) — proof that studying discarding technique pays off in the long run!

- March 2000 (updated May 2001)


 
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